Formula:KLS:14.08:32

<< Formula:KLS:14.08:31

formula in Al-Salam-Chihara

Al-Salam-Chihara >>

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{Q_{n}\!\left(\cos\left(\ln q^{x}+\phi\right);q^{\lambda}{\mathrm{e}^{\mathrm{i}\phi}},q^{\lambda}{\mathrm{e}^{-\mathrm{i}\phi}}\,|\,q\right)}{\left(q;q\right)_{n}}=P^{(\lambda)}_{n}\!\left(x;\phi\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1

Bibliography

Equation in Section 14.8 of KLS.

URL links

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<< Formula:KLS:14.08:31

formula in Al-Salam-Chihara

Al-Salam-Chihara >>